Enhanced matrix function approximation

Abstract

Matrix functions of the form $f(A)v$, where $A$ is a large symmetric matrix, $f$ is afunction, and $v\\ne 0$ is a vector, are commonly approximated by first applying a few,say $n$, steps of the symmetric Lanczos process to $A$ with the initial vector $v$ in order todetermine an orthogonal section of $A$. The latter is represented by a (small)$n\\times n$ tridiagonal matrix to which $f$ is applied. This approach uses the $n$ firstLanczos vectors provided by the Lanczos process. However, $n$ steps of the Lanczosprocess yield $n+1$ Lanczos vectors. This paper discusses how the $(n+1)$stLanczos vector can be used to improve the quality of the computed approximation of$f(A)v$. Also the approximation of expressions of the form $v^Tf(A)v$ is considered.